Proof: This is a \(\mathbb {C}^\ast \) -equivariant version of the result used in the proof of [1]}. As in the proof of [2]}, because the \(\mathbb {C}^\ast \)  action on \(Y\)  is étale-locally linearizable, we can find an étale neighborhood \(S^{\prime }\)  of \(y\)  with a closed \(\mathbb {C}^\ast \) -equivariant embedding into a smooth algebraic space with \(\mathbb {C}^\ast \) -action \(V^{\prime }\) , with \(\dim (V^{\prime })=\dim (T_y Y)\) . According to the proof of the proposition REF , the \(\mathbb {C}^\ast \) -equivariant smooth map \(\phi ^{-1}(S^{\prime })\rightarrow S^{\prime }\)  is étale locally a \(\mathbb {C}^\ast \) -equivariant affine fibration, hence we can find a \(\mathbb {C}^\ast \) -equivariant étale neighborhood \(S\)  of \(y\) , \(\mathbb {C}^\ast \) -equivariant embeddings \(i:R:=\phi ^{-1}(S)\rightarrow U\) , \(j:S\rightarrow V\)  into smooth algebraic spaces with \(\mathbb {C}^\ast \) -action \(U,V\)  with dimension \(\dim (U)=\dim (T_x X)\) , \(\dim (V)=\dim (T_y Y)\) , with a \(\mathbb {C}^\ast \) -equivariant smooth map of relative dimension \(d\)  \(\Phi :U\rightarrow V\)  such that \(\Phi \circ i=j\circ \phi |_R\) . Hence as in the proof of [2]}, up to shrinking \(S,V\)  (and then \(R,U\) ) to a \(\mathbb {C}^\ast \) -invariant open subspace we can find a \(\mathbb {C}^\ast \)  invariant regular function \(g:V\rightarrow \mathbb {C}\)  such that \(\iota _{S,V}(t|_S)=j^{-1}(g)+I_{S,V}^2\) , and \(S={\rm Crit}(g)\) , i.e. \((S,V,g,j)\)  is a \(\mathbb {C}^\ast \) -equivariant critical chart of \(Y\)  near \(y\) . Defining the \(\mathbb {C}^\ast \) -equivariant regular function \(f:=g\circ \Phi :U\rightarrow \mathbb {C}\) , because \(s=\phi ^\star (t)\) , one has \(\iota _{R,U}(s|_R)=i^{-1}(f)+I_{R,U}^2\) . Also \(\Phi \circ i=j\circ \phi |_R\)  and \(\Phi ,\phi \)  are smooth of relative
dimension \(d\) , then \(i(R)=\Phi ^{-1}(j(S))\)  in a neighborhood of \(i(x)\) : taking the union of all these neighborhood, one can shrink \(U\)  (and then \(R\) ) to \(\mathbb {C}^\ast \) -invariant open neighborhood satisfying \(i(R)=\Phi ^{-1}(j(S))\) . Because \(\Phi \)  is smooth, one has:
\({\rm Crit}(f)=\Phi ^{-1}({\rm Crit}(g))=\Phi ^{-1}(j(S))=i(R)\) 
